Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

In this work we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a nonlinear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non-Hermitian linear eigenvalue problem. We review a set of state of the art eigensolvers and discuss strategies to recycle spectral informations from one nonlinear step to the next. More precisely, we consider the Implicitly Restarted Arnoldi method, the Krylov-Schur solver and its block-variant as well as the subspace iteration method with Chebyshev acceleration. On a small test example we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information.Nous considérons la résolution de problèmes aux valeurs propres non-linéaires qui interviennent dans l'analyse de stabilité thermo-acoustique de chambres de combustion de grande taille. Nous introduisons brièvement la modélisation physique qui conduit à un problème aux valeurs propres nonlinéaire qui est résolu par un schéma itératif de type point fixe. Chaque itération non-linéaire nécessite la résolution d'un problème aux valeurs propres complexe non-hermitien. Nous examinons un ensemble de solveurs aux valeurs propres linéaires et discutons des stratégies pour recycler des informations spectrales d'une itération non-linéaire à l'autre. Plus précisèment, nous considérons la méthode d'Arnoldi redémarrée implicitement, le solveur Krylov-Schur et sa variante bloc ainsi que la méthode d'itérations de sous-espaces avec l'accélération de Chebyshev. Sur un exemple de taille modeste, nous étudions la pertinence des différents approches proposées et illustrons sur un exemple industriel de grande taille les performances parallèles des variantes de recyclage les plus efficaces

Spectral recycling strategies for the solution of nonlinear eigenproblems in thermoacoustics

International audienceIn this work we consider the numerical solution of large nonlinear eigenvalue problems that arise in thermoacoustic simulations involved in the stability analysis of large combustion devices. We briefly introduce the physical modeling that leads to a non-linear eigenvalue problem that is solved using a nonlinear fixed point iteration scheme. Each step of this nonlinear method requires the solution of a complex non-Hermitian linear eigenvalue problem. We review a set of state of the art eigensolvers and discuss strategies to recycle spectral informations from one nonlinear step to the next. More precisely, we consider the Implicitly Restarted Arnoldi method, the Krylov-Schur solver and its block-variant as well as the subspace iteration method with Chebyshev acceleration. On a small test example we study the relevance of the different approaches and illustrate on a large industrial test case the performance of the parallel solvers best suited to recycle spectral information

ChASE: Chebyshev Accelerated Subspace iteration Eigensolver for sequences of Hermitian eigenvalue problems

Solving dense Hermitian eigenproblems arranged in a sequence with direct solvers fails to take advantage of those spectral properties which are pertinent to the entire sequence, and not just to the single problem. When such features take the form of correlations between the eigenvectors of consecutive problems, as is the case in many real-world applications, the potential benefit of exploiting them can be substantial. We present ChASE, a modern algorithm and library based on subspace iteration with polynomial acceleration. Novel to ChASE is the computation of the spectral estimates that enter in the filter and an optimization of the polynomial degree which further reduces the necessary FLOPs. ChASE is written in C++ using the modern software engineering concepts which favor a simple integration in application codes and a straightforward portability over heterogeneous platforms. When solving sequences of Hermitian eigenproblems for a portion of their extremal spectrum, ChASE greatly benefits from the sequence's spectral properties and outperforms direct solvers in many scenarios. The library ships with two distinct parallelization schemes, supports execution over distributed GPUs, and it is easily extensible to other parallel computing architectures.Comment: 33 pages. Submitted to ACM TOM

Interpolation-restart strategies for resilient eigensolvers

International audienceThe solution of large eigenproblems is involved in many scientific and engineering applications when for instance, stability analysis is a concern. For large simulation in material physics or thermo-acoustics, the calculation can last for many hours on large parallel platforms. On future large-scale systems, the mean time between failures (MTBF) of the system is expected to decrease so that many faults could occur during the solution of large eigenproblems. Consequently, it becomes critical to design parallel eigensolvers that can survive faults. In that framework, we investigate the relevance of approaches relying on numerical techniques, which might be combined with more classical techniques for real large-scale parallel implementations. Because we focus on numerical remedies we do not consider parallel implementations nor parallel experiments but only numerical experiments. We assume that a separate mechanism ensures the fault detection and that a system layer provides support for setting back the environment (processes,. . .) in a running state. Once the system is in a running state, after a fault, our main objective is to provide robust resilient schemes so that the eigensolver may keep converging in the presence of the fault without restarting the calculation from scratch. For this purpose, we extend the interpolation-restart (IR) strategies initially introduced for the solution of linear systems in a previous work to the solution of eigenproblems in this paper. For a given numerical scheme, the IR strategies consist of extracting relevant spectral information from available data after a fault. After data extraction, a well-selected part of the missing data is regenerated through interpolation strategies to constitute a meaningful input to restart the numerical algorithm. One of the main features of this numerical remedy is that it does not require extra resources, i.e., computational unit or computing time, when no fault occurs. In this paper, we revisit a few state-of-the-art methods for solving large sparse eigenvalue problems namely the Arnoldi methods, subspace iteration methods and the Jacobi-Davidson method, in the light of our IR strategies. For each considered eigensolver, we adapt the IR strategies to regenerate as much spectral information as possible. Through extensive numerical experiments, we study the respective robustness of the resulting resilient schemes with respect to the MTBF and to the amount of data loss via qualitative and quantitative illustrations. 1. Introduction. The computation of eigenpairs (eigenvalues and eigenvectors) of large sparse matrices is involved in many scientific and engineering applications such as when stability analysis is a concern. To name a few, it appears in structural dynamics, thermodynamics, thermo-acoustics, quantum chemistry. With the permanent increase of the computational power of high performance computing (HPC) systems by using a larger and larger number of CPU cores or specialized processing units, HPC applications are increasingly prone to faults. To guarantee fault tolerance, two classes of strategies are required. One for the fault detection and the other for fault correction. Faults such as computational node crashes are obvious to detect while silent faults may be challenging to detect. To cope with silent faults, a duplication strategy is commonly used for fault detection [18, 39] by comparing the outputs, while triple modular redundancy (TMR) is used for fault detection and correction [34, 37]. However, the additional computational resources required by such replication strategies may represent a severe penalty. Instead of replicating computational resources, studies [7, 36] propose a time redundancy model for fault detection. It consists in repeating computation twice on the same resource. The advantage of time redundancy models is the flexibility at application level; software developers can indeed select only a set of critical instructions to protect. Recomputing only some instructions instead of the whole application lowers the time redundancy overhead [25]. In some numerical simulations, data naturally satisfy well defined mathematical properties. These properties can be efficiently exploited for fault detection through a periodical check of the numerical properties during computation [10]. Checkpoint/restart is the most studied fault recovery strategy in the context of HPC systems. The common checkpoint/restart scheme consists in periodically saving data onto a reliable storage device such as a remote disk. When a fault occurs, a rollback is performed to the point of the most recent and consistent checkpoint. According to the implemented checkpoint strategy, all processe

Interpolation-restart strategies for resilient eigensolvers

The solution of large eigenproblems is involved in many scientific and engineering applications when, for instance stability analysis is a concern. For large simulation in material physics or thermo-acoustics, the calculation can last for many hours on large parallel platforms. On future large-scale systems, the time interval between two consecutive faults is forecast to decrease so that many faults could occur during the solution of large eigenproblems. Consequently it becomes critical to design parallel eigensolvers which can survive faults. In that framework, we mainly investigate the relevance of approaches relying on numerical techniques that might be combined with more classical techniques for real large scale parallel implementations. Because we focus on numerical remedies we do not consider parallel implementations nor parallel experiments but only numerical experiments.We assume that a separate mechanism ensures the fault detection and that a system layerprovides support for setting back the environment (processes, \ldots) in a running state.Once the system is in a running state, after a fault, our main objective is to provide robust resilient schemes so that the eigensolver may keep converging through the fault without restarting the calculation from scratch. For this purpose, we extend the interpolation-restart (IR) strategies introduced in a previous work for linear systems. For a given numerical scheme, the IR strategies consist in extracting relevant spectral information from available data after a fault. After data extraction, a well selected part of the missing data is regenerated through interpolation strategies to constitute meaningful input to restart the numerical algorithm. A main feature of this numerical remedy that it does not require extra resources, e.i., computational unit or computing time, when no fault occurs.In this paper, we revisit a few state-of-the-art methods for solving large sparse eigenvalue problems namely the Arnoldi methods, subspace iteration methods and the Jacobi-Davidson method, in the light of our IR strategies. For each considered eigensolver, we adapt the IR strategies to regenerate as much spectral information as possible.Through intensive numerical experiments, we illustrate the qualitative behavior of the resulting schemes when the number of faults and the amount of lost data are varied

Passive Aeroelastic Tailoring

The Passive Aeroelastic Tailoring (PAT) project was tasked with investigating novel methods to achieve passive aeroelastic tailoring on high aspect ratio wings. The goal of the project was to identify structural designs or topologies that can improve performance and/or reduce structural weight for high-aspect ratio wings. This project considered two unique approaches, which were pursued in parallel: through-thickness topology optimization and composite tow-steering